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There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes. So my question is: …

Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . Give …

Well there is indeed a "simple" construction of such a space filling curve. Let $\gamma:[0;1]\rightarrow [0;1]^2$ be a space filling curve. Then one can obtain a space filling curve for $[0;1]^3$ by …

Of course the point has the desired property, but I guess, this is not the space you are looking for. As François said, $C=\{0;1\}^\omega$ and so we get $C^2\cong C$.

If you assume, that $f$ can be written as a power series, lets say $f(x)=\sum_i a_ix^i$ (and the power series converges everywhere absolutely), then one can write down a power series for $f\circ f$, …